Ordinary construction, demolition and household debris is all acceptable. Stockbridge publishes its exclusions in detail, which makes this easy to be precise about.
The city's list of material it will not handle covers incinerator ash, foundry sand, explosives, hospital and biological waste, dead animals, chemicals and radioactive materials, oil sludge, asbestos, septic tank and human waste, sewage, gases, human remains, street sweepings, mining waste and sludge. It also covers hazardous refuse of every kind, naming cleaning fluids, crank case oils, cutting oils, paints, acids, caustics, poisons and drugs.
Sharps, broken glass and sharp metal have their own rule. They have to go into a rigid, puncture-resistant, break-resistant container with a tight lid before they go anywhere near a load.
There is a separate list for commercial containers that will simply not be emptied if they hold the wrong thing: large household or industrial appliances, furniture, tires, wooden crates, logs or limbs over four feet long or four inches thick, bed springs, rock, dirt and concrete blocks, and loose uncontainerized garbage or yard debris. Most of those belong in a different container rather than nowhere, and sorting them out at the start is a five-minute job.
Burning and burying are both out. Stockbridge allows neither except where the Georgia rules for solid waste management specifically permit it.
If something is borderline, one call to your account manager settles it beforehand.